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I have a problem with the difference between complete metric space and a sequentially compact metric space.

For the first one every Cauchy sequence converges inside the space, which is no problem.

But for the last one "every sequence has a convergent subsequence." (-Wiki) And it's here that I get lost.

How does this affect the constraints on the space?

Could someone please try to give me an intuitive explanation?

For [1,9] on the real axis we can take the sequence (1,2,3,4,5,6) as an example. How do we find a convergent subsequence in this one?

Have I missunderstood it all?